The order of roational symmetries of the soccer ball?

Dear Colleagues,

We know that the soccer ball has 60 rotational symmetries and this is computed by the relation |G|=|orbG(i)||stabG(i)|where orbG(i) denotes of the orbit of i in G and stabG(i) is the stabilizer of i in G. In this case we consider the set of pentagons and we find that |orbG(i)|=12 and |stabG(i)|=5 so |G|=60.

The question is what if we consider hexagons instead of pentagons and calculate the order of our group then we find that |G|=120!!!!

The problem is in the 60 degree rotational symmetry about the line through the centers of two opposite hexagonal faces but how?!! In fact I do not know and I need your help.